Few remarks.1. Field is obviously $\pi$-periodic with respect to both $x$ and $y$ and it is singular as $x=(m+\frac{1}{2})\pi$ or $y=(n+\frac{1}{2})\pi$ with $m,n\in \mathbb{Z}$ so one needs to consider only square $\{ -\frac{\pi}{2}<x < \frac{\pi}{2}, -\frac{\pi}{2}<y < \frac{\pi}{2}\}$ where $(0,0)$ is an only equilibrium point.

2. Missing: eigenvectors (so directions of separatrices have not been found.